Vector Algebra
Scalar triple product and determinants
Grade 12
Question:
<p>The value of <equation>\begin{vmatrix} a \cdot p & b \cdot p & c \cdot p \\ a \cdot q & b \cdot q & c \cdot q \\ \cdot & \cdot & \cdot \end{vmatrix}</equation> is</p>
<p>(a) <equation>(p \times q)[a \times b \, b \times c \, c \times a]</equation></p>
<p>(b) <equation>2(p \times q)[a \times b \, b \times c \, c \times a]</equation></p>
<p>(c) <equation>4(p \times q)[a \times b \, b \times c \, c \times a]</equation></p>
<p>(d) <equation>(p \times q)|[a \times b \, b \times c \, c \times a]|</equation></p>
Step-by-Step Solution
Key Concept: The determinant of dot products can be expressed as a scalar triple product involving cross products of the original vectors.
Using properties of scalar triple product and determinants with non-coplanar vectors a, b, c, the determinant evaluates to <equation>2(p \times q)[a \times b \, b \times c \, c \times a]</equation>.
Correct Answer: b